This invention is in the field of simulation, or parameter extraction of characteristics of electrical elements used in the design of printed circuit boards, and solid state integrated circuits.
Parameter extraction, or simulation, of electronic elements has a significant role in the design of modern integrated circuits (IC) operating increasingly at frequencies in the range of hundreds of megahertz. Increasing IC operating frequencies, coupled with reduced, submicron size structures, have made xe2x80x9cfull-wavexe2x80x9d simulation critical for components created within an IC that may be operated near resonance.
As described in the parent application, Ser. No. 08/904,488, incorporated herein by reference in its entirety, historically, capacitive and inductive elements were computed from the geometry of an IC by using general purpose field solvers based on finite-difference or finite-element method. Typical of these tools of the prior art is a requirement for volume and/or area discretization. In this case, solutions had to be computed for large numbers of points descriptive of electric and/or magnetic fields of an element within a device. Using this approach, as frequencies go up, the number of elements requiring solution for a practical xe2x80x9cfull wavexe2x80x9d simulation also goes up resulting in large computation time and memory use for the completion of one simulation.
Another approach in the prior art used simulation tools based on layered media integral equation formulations. These are typically used in the microwave and antenna communities. However, these tools employ direct solution methods which restricts them to small problems. In addition, the formulations that they are based on become ill-conditioned at lower frequencies, resulting in numeric difficulties.
Yet another approach of the prior art is the use of integral equation schemes. An example of this approach is FastCap: A multipole accelerated 3-D capacitance extraction program IEEE Transaction on Computer Aided Design 10(10):1447-1459, November 1991, incorporated herein by reference in its entirety. In general, integral equation schemes work by introducing additional equations to enforce boundary conditions at region interfaces. The introduction of multiple equations for multiple boundary conditions can result in a prohibitive increase in problem size again presenting problems with computation time and memory usage.
Another approach of the prior art to solve parameter extraction problems is the use of layered Green""s functions. These functions have traditionally been used in a 2.5D simulation context where the radiating sources are essentially planar, being confined to infinitely thin sheets. This approach has been popular in the microwave and antenna communities. For these communities, 2.5D modeling of the structures is adequate because generally conductor thickness is much smaller than the width. However, in IC and packaging contexts planar modeling is generally insufficiently accurate. Shrinking IC geometry size approaching submicron dimensions dictates that thickness of conductors within an IC is often on the same order as the width. This physical characteristic of internal IC structures reduces the applicability of a strictly planar oriented approach by introducing substantial errors.
Above listed problems are avoided in accordance with one aspect of the invention by an apparatus simulating a component, where the component is conducting a current density. The apparatus has means for discretizing the component into a plurality of triangular elements, a means for computing Green""s function descriptive of the relationship between the elements, a means for computing basis functions relating to said elements, where the basis functions decompose the current density into divergence free and curl free parts, and means for combining the Green""s functions and the basis functions to arrive at the solution to the integral equation representative of the component to be simulated.
The basis functions are computed from rooftop functions formed from the elements. The basis functions, b, constructed using spanning tree T, rooftop functions, h, and triangular elements, t, are used to compute:
Ohmic interactions among the rooftop functions in an hxc3x97h sparse matrix xcexa9;
vector potential interactions among the rooftops functions in an hxc3x97h dense matrix A;
scalar potential interactions among said rooftop functions in an txc3x97t dense matrix "PHgr";
a hxc3x97b sparse matrix V;
and a txc3x97b sparse matrix S descriptive of the divergence of each of the basis function b to express a matrix B representative of the interaction between the basis functions within the component, as represented by
B=VT(xe2x88x92xcexa9xe2x88x92jxcfx89A)Vxe2x88x92ST"PHgr"S
A preconditioner P is used to compute an approximation to the inverse of the resulting B matrix
P=VT(xe2x88x92xcexa9xe2x88x92jxcfx89Ã)Vxe2x88x92ST{tilde over ("PHgr")}S
where {tilde over ("PHgr")} contains the self interactions among said rooftop functions, and Ã contains the interactions among the rooftop functions and interactions between the rooftops that share one of the triangular elements.
The inverse of B, containing the necessary information for computing parameters of the component of interest is computed by using the sparse preconditioner P as an approximation to be used iteratively to solve:
Pxe2x88x921Bx=Pxe2x88x921s
where s is a stimulus.